Consider the Lie algebra $\text{sl}(2,\mathbb{C})$ (commutator as Lie bracket), with its standard basis consisting of $$e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$
Define the map $R: \text{sl}(2,\mathbb{C})\rightarrow \text{End}(\mathbb{C}[x,y])$ by $$h\mapsto x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}, \quad e\mapsto x\frac{\partial}{\partial y}, \quad f\mapsto y\frac{\partial}{\partial x}.$$
This realizes $(\mathbb{C}[x,y],R)$ as a representation of $\text{sl}(2,\mathbb{C})$.
I want to know whether this representation is irreducible, i.e., whether there is an invariant vector subspace of $\mathbb{C}[x,y]$ which is not equal to $0$ or $\mathbb{C}[x,y]$.
Hint Each of the basis operators preserve the total degree of any polynomial (or map that polynomial to zero).